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The complexity of density

By Joshua Arbury, on January 26th, 2012

You would think that calculating, and analysing, the density of a city would be a fairly perfunctory mathematical task, and would tell us useful information about the nature of that city. As I noted in this previous blog post, perhaps the most challenging aspect of calculating a city’s ‘average density’ is working out where its boundaries are. For Auckland, there are a variety of boundaries and therefore a variety of average densities. But even that approach can lead to some surprising results – as pointed out in Paul Mees’s book “Transport for Suburbia”, New York and Los Angeles have a similar density, so do Vancouver and Las Vegas. Yet each city has a significantly different level of public transport usage. Mees puts this down to the quality of public transport provision being more important than density when it comes to ridership, although equally one could also start asking questions about how we’ve measured density – especially as saying Los Angeles and New York have the same density just seems to be so incorrect.

The issue of density is looked at in detail in this excellent article by Eric Eidlin, a community planner and Sustainable Communities Partnership Liaison for the Federal Transit Administration in San Francisco. He questions whether average density over the whole metropolitan area is really a particularly useful figure when assessing how ‘dense’ or ‘sprawled’ in reality an urban area is. He presents the conundrum we face when looking at density:

Many people …tend to think of “sprawling” cities as places where people make most of their trips by car, and non-sprawling cities as places where people are more likely to walk, cycle, or take transit. This is why Los Angeles, which has more vehicles per square mile than any other urbanized area, and where transit accounts for only two percent of the region’s overall trips, is considered sprawling, while the New York urbanized area is not. We also know (or think we know) that places where people frequently walk, cycle, or take transit tend to have high population densities, and for this reason we tend to view low density as a proxy for sprawl. But as it turns out, the Los Angeles urbanized area—which in both myth and fact is very car-oriented—is also very dense. In fact, Los Angeles has been the densest urbanized area in the United States since the 1980s, denser even than New York and San Francisco.

These facts present a bit of a mystery. If one were to measure sprawl by measuring a region’s average level of Vehicle Miles Traveled (VMT), Los Angeles would certainly qualify as sprawling. But if we measure sprawl by population density, LA would not sprawl at all. In fact, it would be the least sprawling urbanized area in the country. How can Los Angeles be so dense and yet also exhibit so many characteristics associated with sprawl, including high levels of car travel (both in per capita and absolute terms) and low rates of walking, bicycling and transit ridership?

A useful way we can start deconstructing the issue of density is to think of two A4 pieces of paper, each with 100 dots on it. On one piece of paper each dot is equally spaced, while on the other piece of paper the dots cluster together in places and are very widely spaced in other places.

Overall, both pieces of paper have the same average density of dots. But really, their distribution is very different. If each dot was to represent 100 people and the paper represented a city, you would have vastly different cities even though their overall density is the same. This is explained in the article:

Sprawl is a regional attribute, so when observers point out that LA is denser than New York, they are not talking about the cities of Los Angeles and New York. Rather they are talking about the urbanized area, which is essentially the combined area of the cities and their suburbs. The other part of the answer is that density by itself—the simple ratio of population to square mile—is not a very useful way to measure sprawl. What matters is the distribution of density, or how evenly or unevenly an area’s population is spread out across its geographic area. If we look at the density distribution in Los Angeles, we notice that its suburbs are much denser than those of other large U.S. cities, such as New York, San Francisco or Chicago. These high-density suburbs compensate for the comparatively low density of LA’s urban core, and, in so doing, increase the average density of the area as a whole. In other words, Los Angeles has both a relatively high density and a relatively even distribution of density throughout its urbanized area.

So, if we continue to use our “dots on a piece of paper” example, Los Angeles would be much closer to the evenly spaced dots example, whereas New York has a huge concentration of dots in its inner area (Manhattan, Bronx, Brooklyn and parts of Queens in particular) and then quite widely spaced dots further out (outer Long Island, north into New York state and west into New Jersey).

It’s pretty clear then that ‘average density’ over a whole urban area doesn’t really tell us too much about the characteristics of that urban area. But how might we examine density in a more helpful way? The article, thankfully, provides us with some options:

One approach is to measure the extent to which the population density varies across an urban area. Using a statistical tool called the Gini coefficient, we can get a sense of the degree of variation for different urban areas. The Gini coefficient is based on the Lorenz curve, a cumulative frequency curve that compares the distribution of a specific variable (in this case, population density) with a uniform distribution that represents perfect equality.

Using such a measure to compare Los Angeles, New York and San Francisco gives us the following results:

You can see, in particular, how much more of New York and San Francisco’s population is concentrated in a small proportion of land area than in the case for Los Angeles. This is detailed further:

In Los Angeles, 40 percent of the population live on the most densely settled 10 percent of land. By way of comparison, roughly 66 percent of New York’s population, and 67 percent of San Francisco’s, live on the most densely settled ten percent of the land. By looking even further to the right of the graph, one finds that 25 percent of the population in Los Angeles lives on the densest 5 percent of the land. By contrast, 46 percent of San Francisco’s population, and more than 50 percent of New York’s, live on the densest 5 percent of the land. The overwhelming majority of New York and San Francisco’s residents live on a very small portion of their urbanized areas’ land. But this is much less the case in LA.

A second way of measuring density more helpfully is through what’s called ‘perceived density’. This weights the density of an area by the proportion of the area’s population that lives there, effectively measuring the average number of people around each resident of the city. The method of calculation is helpfully described by the example of the fictional city of Metropolis:

Metropolis has a central core of 100,000 residents who live on ten square miles of land and a suburb with 10,000 residents who live on 100 square miles of land. The standard density of Metropolis is 1,000 people per square mile. However, since 90 percent of the population—those who inhabit the core—live in a very dense environment, this standard density number has little bearing on the way most residents experience their city. By giving the core’s density a weight of 90 percent and the suburb’s density a weight of 10 percent—weights that are equal to the respective proportions of the city’s residents that inhabit each part—we get an adjusted density of 9,100 people per square mile, a number that more closely approximates the density at which the average resident of Metropolis lives.

Comparing the different US cities under this ‘perceived density’ measure gives the following results: Under this measurement system we see New York really standing out from any other city in the USA. Los Angeles is still fairly high up there though, in third place. And our comparison with public transport and walking – while better than average density – still doesn’t exactly align up perfectly.

A third measurement system is also included in the table above, the density gradient index. This is described below:

Bradford pushed the concept of perceived density a step further by developing the density gradient index. The density gradient index, which is the ratio of perceived density to standard density, is an indication of the unevenness of population distribution—or, to use Bradford’s terminology—a measure of “clumpiness.” Table 2 also shows the density gradient index for each urbanized area.

Overall, when comparing the different measurements of density with public transport use, we get the following:

Bradford did a regression analysis to analyze the relationship between perceived density and commute mode (the final two columns of Table 2). He found virtually no association between standard density and the percentage of workers commuting by public transit or walking, but a strong association between perceived density and commuting by transit or foot, and an even stronger association between the density gradient index and the percentage of workers commuting by transit or by foot.

What does that mean for Auckland? Well, until we can analyse our population distribution, perceived density and density gradient index, who knows whether we’re really at the same urban density as Sydney – like the average density statistics will tell us. The best graph I have seen so far is included below, and suggests that perhaps Auckland’s density is a little bit “Los Angeles” compared to Sydney’s “New York”. It would seem that if we’re thinking about land-use policies to boost public transport, walking (and presumably cycling) use, then it may be useful for the “lumpiness” of Auckland’s population density to increase – obviously particularly around our rapid transit network. Fortunately, that’s what most of our plans seem to propose.

16 comments to The complexity of density

Great analysis. I want to be able to work and recreate near to wear I live. I don’t want to have to travel vast distances to be able to do this. I also want to live within cycling distance to a transport hub. I think these criteria would make Auckland more ‘livable’.

Another great post, admin. That last chart is fascinating; so Ak’s bump around 40 people/ha are the older inner tram-built suburbs? And look at Perth! those who claim that AK is not dense enough for PT to work have got a huge problem with Perth. If investment in rail integrated with feeder buses and park and rides/kiss and rides can make a place with Perth’s spread of population become such a huge user of PT image what we could achieve in Auckland with its already much tighter population.

Clearly encouraging more development around transport nodes is the smart thing to do but it is also clear that the real key is in providing a smart and modern system in the first place. Which, anyway, is a great way to encourage growth around those nodes. Arguing that the density of AK must reach Tokyo levels before we need real transit is a way of killing investment by delay. Here is a story on the upkick in the use of NY’s PATH; highest ridership in 50 years. Why? Investments in improvements:http://blogs.wsj.com/metropolis/2012/01/24/path-sees-spike-in-trans-hudson-commuters/?mod=WSJBlog&mod=WSJ_NY_NY_Blog

Density levels are fraught with danger. After all Hong Kong would be far less dense if you included the uninhabitable parts of Hong Kong Island and the new territories. Similarly with Wellington, or Auckland for that matter. Do you include the harbours? After all no one can live there, yet they cause the city to spread out more because they are uninhabitable.

Surely physical geography is also a significant factor. New York and San Francisco / Oakland both have high patronage rates and both have geography dominated by significant bodies of water. This water constrains the number and direction of roads approaching the centre, such that a relatively small number of PT corridors are much less disadvantaged than would be the case if the road network penetrated the centre from any direction and commuters could travel in pretty much a straight line from any point of the compass.

Geography is surely a significant factor in Auckland – much more so than in say Perth. Yet with this argument you might have expected a greater level of patronage in Auckland. Have we perhaps not allowed PT to take advantage of the constrained corridors?

Now those who lived and prospered throught the 50’s and 60’s in New Zealand will not want to hear this but I am hearing more and more positive comments about Auckland as a great city to live and work in. Mostly these comments come from Kiwi’s who have returned from their extended oe’s or recent arrivals from Europe. As we focus more and more on intensification rather than the old ‘greenbelt’ development approach, Auckland will become more like London and New York. These are actually cities that people want to live and work in. That will be one way to stop the dreaded ‘brain drain’ to Aussie and the UK. Part of this is due to developments like Wynyard and the dramatic improvements in PT over the past 10 years.

Auckland is up near the top of all the livability rankings discussed in http://en.wikipedia.org/wiki/Livability_ranking …All the rankings are subjective depending on the factors the ranker feels are important, but Auckland is held to be a better place than just about anywhere in Europe. Having lived all over Europe, I’d agree.

Lets do the same for Sydney: for densities 5-60 we have percentages 7 6 7 8 12 13 12 8 5 3 2 2, for densities 65-115 we have 1 percent and densities of 120-150, 0.5%. Then we have a perceived density for Sydney of 38.6 people per ha, or 9990 people per square mile….

Yes, it is surprising given what you usually hear. But looking at yr graph above, it is not that surprising — all the perceived density is is the mean value (the mid-point) of the density distribution. Sydney’s is pulled down by all that weight on the left (large numbers of people at quite low densities) but dragged up by the very high densities on the right. While Auckland also has a heavy left tail, it is not as bad as Sydney’s and it is far more peaked in the 30-45p/ha zone…

What’s quite interesting is the lack of people in the 60-80 persons/ha zone. Is it that it just isn’t viable for developers to build at that density? What would properties with that kind of density look like? perhaps something like 2-3 story apartment buildings which is something that isn’t over the top and won’t dominate and shade neighbourhoods.

I think that if we could shift the top of the hump up by around 10 people/ha then it would make a huge difference to the density while not crowding out the city with high rise’s that those that are promoting sprawl like to suggest.

Nice deduction Matt, and looking at the plan, feeding those blocks into the corridors like at Tamaki, New Lynn, Gt Nth Rd Grey Lynn, that could be achieved and Mr Quax could still sleep at night out in ‘burbs without even knowing about it. zzzzzzzz.

I just stumbled on this blog. Regarding what 60-80/ha looks like, it doesn’t look very dense. I live in Toronto and there are single family census tracts in our suburbs with that kind of density. They’re often not just detached homes, but also semi-detached and maybe a few townhomes. These suburban areas are still very, well, suburban, with substantial setbacks and relative wide (~8m) streets. If you were to have townhouses like those of Centre City Philadelphia, you could probably hit densities of 150/ha. With 2-3 storey apartments like in much of inner city Montreal, you could hit densities of 200/ha. The highest density you could achieve with highrises is probably around 1000/ha, maybe even more.